Finite Group Databasebeta

← #46822722

Subgroups — #46822722

Order 34432.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771295C211yesmaximal, characteristic, complemented
2#31771251C211yesmaximal, characteristic
3#31770960C211yesmaximal, characteristic, complemented
4#31770936C211yesmaximal, characteristic, complemented
5#31770930C211yesmaximal, characteristic, complemented
6#31770906C211yesmaximal, characteristic, complemented
7#31770846C211yesmaximal, characteristic, complemented
8#31770834C211yesmaximal, characteristic, complemented
9#31770828C211yesmaximal, characteristic, complemented
10#31770769C211yesmaximal, characteristic, complemented
11#31770758C211yesmaximal, characteristic, complemented
12C4×#16864391C211yesmaximal, characteristic, complemented
13#31770602C211yesmaximal, characteristic, complemented
14C2×#22575375C211yesmaximal, characteristic, complemented
15C8×D4×C269C211yesmaximal, characteristic, complemented
16C2×#16864402C2211yescharacteristic
17C22×D1076C2211yescharacteristic
18C2×Dic1076C2211yescharacteristic, complemented
19C2×#16864391C2211yescharacteristic
20C2×#16864391C2211yescharacteristic
21#22575446—21no—
22#22575445C2211yescharacteristic, complemented
23#22575443—21no—
24#22575441C2211yescharacteristic
25#22575438—21no—
26#22575434—21no—
27#22575432C2211yescharacteristic
28#22575432C2211yescharacteristic
29#22575432C2211yescharacteristic, complemented
30#22575427C2211yescharacteristic
31#22575426—21no—
32#22575421—21no—
33C4×Dic538C2211yescharacteristic, complemented
34C4×D1076C2211yescharacteristic
35#22575412C2211yescharacteristic, complemented
36#22575409—21no—
37C2×C4⋊Dic269C2211yescharacteristic
38C22⋊C4×D269—21no—
39#22575387C2211yescharacteristic, complemented
40#22575385C2211yescharacteristic, complemented
41#22575385C2211yescharacteristic, complemented
42#22575378C2211yescharacteristic, complemented
43#22575378C2211yescharacteristic, complemented
44#22575378C2211yescharacteristic, complemented
45#22575378C2211yescharacteristic, complemented
46#22575375C2211yescharacteristic
47#22575375C2211yescharacteristic
48#22575375C2211yescharacteristic
49#22575375C2211yescharacteristic
50#22575375C2211yescharacteristic
51#22575375C2211yescharacteristic
52C22×C8×C269C2211yescharacteristic
53C22×C8×C269C2211yescharacteristic, complemented
54C4×D4×C269C2211yescharacteristic
55C4⋊C8×C269C2211yescharacteristic, complemented
56C22⋊C8×C269C2211yescharacteristic
57C22⋊C8×C269C2211yescharacteristic, complemented
58C4×C8×C269C2211yescharacteristic, complemented
59C23×D269—21no—
60#16864402—22no—
61#16864402—41no—
62C2×Dic538C2311yescharacteristic
63C2×Dic538C2311yescharacteristic
64C2×D1076—22no—
65C2×D1076—21no—
66C2×D1076—41no—
67C2×D1076C2311yescharacteristic
68C2×D1076C2311yescharacteristic
69C2×C22⋊D269—21no—
70C2×C22⋊D269—21no—
71C22×Dic269—21no—
72C2×C4×D269—21no—
73C2×C4×D269—21no—
74Dic1076—41no—
75#16864391—22no—
76#16864391—41no—
77C4⋊Dic269—21no—
78C4⋊Dic269C2311yescharacteristic
79C4⋊Dic269C2311yescharacteristic
80C4⋊Dic269C2311yescharacteristic
81C4⋊Dic269C2311yescharacteristic
82#16864381—21no—
83#16864381—21no—
84#16864381—21no—
85#16864381—21no—
86#16864381—21no—
87#16864381—21no—
88C22⋊Dic269—21no—
89C22⋊Dic269—21no—
90C22⋊Dic269—21no—
91C22⋊Dic269—21no—
92#16864379—21no—
93#16864379—21no—
94#16864379—21no—
95#16864379—21no—
96C4×Dic269—21no—
97C4×Dic269—21no—
98C2×D4×C269D411yescharacteristic
99C22×C4×C269C2311yescharacteristic
100C22×C4×C269C2311yescharacteristic
101C2×C8×C269—41no—
102C2×C8×C269—41no—
103C2×C8×C269C2311yescharacteristic
104C2×C8×C269C2311yescharacteristic
105C2×C8×C269C2311yescharacteristic
106C2×C8×C269C2311yescharacteristic
107C4⋊C4×C269D411yescharacteristic
108C22⋊C4×C269D411yescharacteristic
109C22⋊C4×C269D411yescharacteristic
110C42×C269C2311yescharacteristic
111C22×D269—21no—
112C22×D269—21no—
113C22×D269—42no—
114C22×D269—42no—
115C22×D269—81no—
116Dic538—21no—
117Dic538—21no—
118Dic538—21no—
119Dic538—21no—
120D1076—21no—
121D1076—21no—
122D1076—21no—
123D1076—42no—
124D1076#4112yes—
125C22⋊D269—42no—
126C22⋊D269—81no—
127C2×Dic269—21no—
128C2×Dic269—21no—
129C2×Dic269—21no—
130C2×Dic269—21no—
131C2×Dic269—21no—
132C2×Dic269—21no—
133C2×Dic269—42no—
134C4×D269—42no—
135C4×D269—81no—
136C23×C269C2×D411yescharacteristic
137C23×C269C2×D411yescharacteristic
138D4×C269—22no—
139C2×C4×C269—21no—
140C2×C4×C269—21no—
141C2×C4×C269—21no—
142C2×C4×C269—21no—
143C2×C4×C269C2×D411yescharacteristic
144C2×C4×C269C2×D411yescharacteristic
145C2×C4×C269C2×D411yescharacteristic
146C2×C4×C269C2×D411yescharacteristic
147C2×C4×C269C2411yescharacteristic
148C8×C269—21no—
149C8×C269—21no—
150C8×C269—21no—
151C8×C269—21no—
152C22×C269—21no—
153C22×C269—21no—
154C22×C269—21no—
155C22×C269—21no—
156C22×C269—21no—
157C22×C269#13512yes—
158C22×C269C22×D411yescharacteristic
159C2×D269—22no—
160C2×D269—41no—
161C2×D269—41no—
162C2×D269—41no—
163C2×D269—41no—
164C2×D269—41no—
165C2×D269—84no—
166C4×C269—21no—
167C4×C269—21no—
168C4×C269—21no—
169C4×C269—21no—
170C4×C269C2×#4111yescharacteristic
171C4×C269H(5,𝔽2)11yescharacteristic
172Dic269—41no—
173Dic269—41no—
174Dic269—41no—
175Dic269—41no—
176Dic269—41no—
177Dic269—41no—
178C2×C269—22no—
179C2×C269—41no—
180C2×C269#54611yescharacteristic
181C2×C269C2×#13511yescharacteristic
182C2×C269#57711yescharacteristic
183D269—42no—
184D269—81no—
185C269#329611yescharacteristic, complemented, Sylow
186#3296—2691nomaximal, Sylow
187#552—2691no—
188#546—2691no—
189#488—2691no—
190#483—2691no—
191#482—2691no—
192#478—2691no—
193#467—2691no—
194#465—2691no—
195#464—2691no—
196#452—2691no—
197#450—2691no—
198C4×SD16—2691no—
199C8×D4D26911yescharacteristic, complemented
200#416—2691no—
201C2×#102—2691no—
202C2×#41—2691no—
203C22×D4—2691no—
204C2×Q16—2691no—
205C2×SD16—2691no—
206C2×SD16—2691no—
207C22×C8C2×D26911yescharacteristic
208C22×C8C2×D26911yescharacteristic, complemented
209#126—5381no—
210#125—2691no—
211#125—5381no—
212#124—2691no—
213#123—5381no—
214#123—5381no—
215C22⋊Q8—2691no—
216C22⋊Q8—2691no—
217C22⋊Q8—2691no—
218#121—2691no—
219#121—5381no—
220#120—5381no—
221C4×Q8—2691no—
222C4×D4—2691no—
223C4×D4C2×D26911yescharacteristic
224#117—2691no—
225#117—5381no—
226C2×C4⋊C4—2691no—
227C2×C22⋊C4—5381no—
228#107—2691no—
229#106—2691no—
230#106—2691no—
231C4⋊C8C2×D26911yescharacteristic, complemented
232#103—2691no—
233#103—2691no—
234#103—2691no—
235#103—2691no—
236#102—2691no—
237#102—2691no—
238#102—2691no—
239#102—2691no—
240#102—2691no—
241#102—2691no—
242C22⋊C8C2×D26911yescharacteristic
243C22⋊C8C2×D26911yescharacteristic, complemented
244C4×C8C2×D26911yescharacteristic, complemented
245C24—5381no—
246#41—5382no—
247#41—10761no—
248C2×Q8—2691no—
249C2×Q8—2691no—
250C2×D4—2691no—
251C2×D4—2691no—
252C2×D4—5382no—
253C2×D4—5381no—
254C2×D4—5381no—
255C2×D4—5381no—
256C2×D4—10761no—
257C2×D4D107611yescharacteristic
258C22×C4—5381no—
259C22×C4—5381no—
260C22×C4—5381no—
261C22×C4C22×D26911yescharacteristic
262C22×C4C22×D26911yescharacteristic
263Q16—10761no—
264SD16—5382no—
265SD16—10761no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8C22×D26911yescharacteristic
269C2×C8C22×D26911yescharacteristic
270C2×C8C22×D26911yescharacteristic
271C2×C8C22×D26911yescharacteristic
272C4⋊C4—2691no—
273C4⋊C4—2691no—
274C4⋊C4—2691no—
275C4⋊C4—2691no—
276C4⋊C4—5381no—
277C4⋊C4—5381no—
278C4⋊C4—5381no—
279C4⋊C4—5381no—
280C4⋊C4—5381no—
281C4⋊C4—5381no—
282C4⋊C4—5381no—
283C4⋊C4D107611yescharacteristic
284C22⋊C4—5381no—
285C22⋊C4—5381no—
286C22⋊C4—5381no—
287C22⋊C4—5381no—
288C22⋊C4—5381no—
289C22⋊C4—5381no—
290C22⋊C4—5381no—
291C22⋊C4—5381no—
292C22⋊C4D107611yescharacteristic
293C22⋊C4D107611yescharacteristic
294C42—5381no—
295C42—5381no—
296C42C22×D26911yescharacteristic
297C23—5381no—
298C23—5381no—
299C23—10762no—
300C23—10762no—
301C23—21521no—
302C23C2×D107611yescharacteristic
303C23C2×D107611yescharacteristic
304Q8—5381no—
305Q8—5381no—
306Q8—5381no—
307Q8—5381no—
308D4—22no—
309D4—2692no—
310D4—5381no—
311D4—5381no—
312D4—5381no—
313D4—10762no—
314D4—10762no—
315D4—21521no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—5381no—
321C2×C4—5381no—
322C2×C4—5381no—
323C2×C4—5381no—
324C2×C4—5381no—
325C2×C4—5381no—
326C2×C4—10762no—
327C2×C4—10762no—
328C2×C4—21521no—
329C2×C4C2×D107611yescharacteristic
330C2×C4C2×D107611yescharacteristic
331C2×C4C2×D107611yescharacteristic
332C2×C4C2×D107611yescharacteristic
333C2×C4C23×D26911yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—5382no—
344C22—10761no—
345C22—10761no—
346C22—10761no—
347C22—10761no—
348C22—10761no—
349C22—21524no—
350C22#2257547512yes—
351C22C22×D107611yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—10761no—
357C4—10761no—
358C4—10761no—
359C4—10761no—
360C4—10761no—
361C4—10761no—
362C4#41×D26911yescharacteristic
363C4#2257549811yescharacteristic
364C2—22no—
365C2—41no—
366C2—10762no—
367C2—21521no—
368C2#3177125011yescharacteristic
369C2C2×#2257547511yescharacteristic
370C2#3177141711yescharacteristic
subgroupG/H
centerC22C22×D1076
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC8×D4×C269C2
socleC22×C269C22×D4
prime radical#31771251C2
hypercenterC8×D4D269
hypocenterC269#3296
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #31771295C2semidirect
3. #31770960C2semidirect
4. #31770936C2semidirect
5. #31770930C2semidirect
6. #31770906C2semidirect
7. #31770846C2semidirect
8. #31770834C2semidirect
9. #31770828C2semidirect
10. #31770769C2semidirect
11. #31770758C2semidirect
12. C4×#16864391C2semidirect
13. #31770602C2semidirect
14. C2×#22575375C2semidirect
15. C8×D4×C269C2semidirect
18. C2×Dic1076C22semidirect
22. #22575445C22semidirect
29. #22575432C22semidirect
33. C4×Dic538C22semidirect
35. #22575412C22semidirect
39. #22575387C22semidirect
40. #22575385C22semidirect
41. #22575385C22semidirect
42. #22575378C22semidirect
43. #22575378C22semidirect
44. #22575378C22semidirect
45. #22575378C22semidirect
53. C22×C8×C269C22semidirect
55. C4⋊C8×C269C22semidirect
57. C22⋊C8×C269C22semidirect
58. C4×C8×C269C22semidirect
185. C269#3296semidirect
199. C8×D4D269semidirect
208. C22×C8C2×D269semidirect
231. C4⋊C8C2×D269semidirect
243. C22⋊C8C2×D269semidirect
244. C4×C8C2×D269semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#317712952
#317712512
#317709602
#317709362
#317709302
#317709062
#317708462
#317708342
#317708282
#317707692
#317707582
C4×#168643912
#317706022
C2×#225753752
C8×D4×C2692
#3296269